On the classgroup of integral grouprings of finite abelian groups

In this note I settle a question which arose out of my first paper under the above title ( cf . [1]), where I considered the classgroup C (Z(Γ)) of the integral groupring Z(Γ) of a finite Abelian group Γ. This classgroup maps onto the classgroup C( ) of the maximal order of the rational groupring Q (Γ), and C ( ) is the product of the ideal classgroups of the algebraic number fields which occur as components of Q (Γ) and is thus in a sense known. One is then interested in the kernel D (Z(Γ)) of C (Z(Γ)) → C ( ) and in its order k (Γ). In [1] I proved that, for Γ a p -group, k (Γ) is a power of p . I also computed k (Γ) for small exponents. My computation used crucially the fact that, for the groups Γ considered, the groups of units of algebraic integers which occurred were finite, i.e. that the only number fields which turned up were Q and Q ( n ) with n 4 = 1 or n 6 = 1. The numerical results obtained led me to the question whether in fact k (Γ) tends to infinity with the order of Γ.